Locomotion with a wavy cylindrical filament in a yield-stress fluid

A yield stress is added to Taylor’s (1952, Proc. Royal Soc. A, 211, 225-239) model of a microscopic organism with a wavy cylindrical tail swimming through a viscous fluid. Viscoplastic slender-body theory is employed for the task, generalising existing results for Bingham fluid to the Herschel–Bulkley constitutive model. Numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength, and a Bingham number which describes the strength of the yield stress. Numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. If the wave amplitude is sufficiently small, the yield stress of the material inevitably dominates the flow; the resulting ‘plastic locomotion’ results in swimming speeds that depend strongly on the swimming gait, and can, in some cases, even be negative. Conversely, when the yield stress is large, swimming becomes possible at the wave speed, with the swimmer sliding or burrowing along its centreline with a relatively high efficiency

Taylor's swimming sheet in a yield-stress fluid

A yield stress is added to Taylor’s (Proc. R. Soc. Lond. A, vol. 209, 1951, pp. 447–461) model of a two-dimensional flexible sheet swimming through a viscous fluid. Both transverse waves along the sheet, as in Taylor’s original model, and longitudinal waves are considered as means of locomotion. In each case, numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength and a Bingham number which describes the strength of the yield stress. The numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. When the yield stress is large, the swimming speed for low wave amplitude is exactly double that for a Newtonian fluid, for either type of wave

Viscoplastic slender-body theory

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid

Building on Oldroyd’s viscoplastic legacy: Perspectives and new developments

The decade following the second world war heralded the publication of a collection of important papers on non-Newtonian fluid mechanics; Oldroyd’s work featured heavily in this collection. Not only did these articles establish important results, but Oldroyd’s style and methods set the scene for subsequent work in the area, exploiting mathematical analysis to formulate problems, establish results and guide further research. While Oldroyd’s name will forever be linked with the study of elastic fluids, the purpose of the present paper is to offer a modern perspective on a number of Oldroyd’s papers on viscoplastic fluids from 1947–1951 [1], [2], [3], [4], [5], [6], [7], [8]. Along the way, we sprinkle in a brief review of some of the subsequent developments stemming from Oldroyd’s advances, together with a few new results guided by his work. Following the approach of most of Oldroyd’s original papers, we focus on unidirectional flow down conduits. In an Appendix, we complement this discussion with a lubrication analysis, extending, clarifying and correcting the important original analysis of Walton and Bittleston (1991) [9]; although lubrication theory was not directly utilized by Oldroyd, the methodology aligns with his philosophy of using asymptotic and analytical approaches

Translating and squirming cylinders in a viscoplastic fluid

Three related problems of viscoplastic flow around cylinders are considered. First, translating cylinders with no-slip surfaces appear to generate adjacent rotating plugs in the limit where the translation speed becomes vanishingly small. In this plastic limit, analytical results are available from plasticity theory (slipline theory) which indicate that no such plugs should exist. Using a combination of numerical computations and asymptotic analysis, we show that the plugs of the viscoplastic theory actually disappear in the plastic limit, albeit very slowly. Second, when the boundary condition on the cylinder is replaced by one that permits sliding, the plastic limit corresponds to a partially rough cylinder. In this case, no plasticity solution has been previously established; we provide evidence from numerical computations and slipline theory that a previously proposed upper bound (Martin & Randolph, Geotechnique, vol. 56, 2006, pp. 141–145) is actually the true plastic solution. Third, we consider how a prescribed surface velocity field can propel cylindrical squirmers through a viscoplastic fluid. We determine swimming speeds and contrast the results with those from the corresponding Newtonian problem

Obstructed and channelized viscoplastic flow in a Hele-Shaw cell

A theoretical study is presented of the flow of viscoplastic fluid through a Hele-Shaw cell that contains various kinds of obstructions. Circular and elliptical blockages of the cell are considered together with stepwise contractions or expansions in slot width, all within the simplifying approximation of a narrow gap. Specific attention is paid to the flow patterns that develop around the obstacles, particularly any stagnant plugged regions, and the asymptotic limits of relatively small or large yield stress. Periodic arrays of circular contractions or expansions are studied to explore the interference between obstructions. Finally, viscoplastic flow through a cell with randomly roughened walls is examined, and it is shown that constructive interference of local contractions and expansions leads to a pronounced channelization of the flow. An optimization algorithm based on minimization of the pressure drop is derived to construct the path of the channels in the limit of relatively large yield stress or, equivalently, relatively slow flow.D.R.H. is grateful to the Killam Foundation for a Postdoctoral Fellowship.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.

Obstructed and channelized viscoplastic flow in a Hele-Shaw cell

A theoretical study is presented of the flow of viscoplastic fluid through a Hele-Shaw cell that contains various kinds of obstructions. Circular and elliptical blockages of the cell are considered together with stepwise contractions or expansions in slot width, all within the simplifying approximation of a narrow gap. Specific attention is paid to the flow patterns that develop around the obstacles, particularly any stagnant plugged regions, and the asymptotic limits of relatively small or large yield stress. Periodic arrays of circular contractions or expansions are studied to explore the interference between obstructions. Finally, viscoplastic flow through a cell with randomly roughened walls is examined, and it is shown that constructive interference of local contractions and expansions leads to a pronounced channelization of the flow. An optimization algorithm based on minimization of the pressure drop is derived to construct the path of the channels in the limit of relatively large yield stress or, equivalently, relatively slow flow.D.R.H. is grateful to the Killam Foundation for a Postdoctoral Fellowship.This is the author accepted manuscript. The final version is available from Cambridge University Press via http://dx.doi.org/10.1017/jfm.2016.

Dewatering of fibre suspensions by pressure filtration

A theoretical and experimental study of dewatering of fibre suspensions by uniaxial compression is presented. Solutions of a one-dimensional model are discussed and asymptotic limits of fast and slow compression are explored. Particular focus is given to relatively rapid compression and to the corresponding development of spatial variations in the solidity and velocity profiles of the suspension. The results of complementary laboratory experiments are presented for nylon or cellulose fibres suspended in viscous fluid. The constitutive relationships for each suspension were measured independently. Measurements of the load for different fixed compression speeds, together with some direct measurements of the velocity profiles using particle tracking velocimetry, are compared with model predictions. The comparison is reasonable for nylon, but poor for cellulose fibres. An extension to the model, which allows for a strain-rate-dependent component in the network stress, is proposed, and is found to give a dramatic improvement in the model predictions for cellulose fibre suspensions. The reason for this improvement is attributed to the microstructure of cellulose fibres, which, unlike nylon fibres, are themselves porous.Akzo Nobel, Valmet Ltd., Natural Sciences and Engineering Council of Canada, Killam Postdoctoral Fellowshi